If you had zero other income in the second state, you could be justified in claiming all of the interest income to your first state if for no other reason than it is easier to do so. In addition, if the amount is small, it won't have any impact at all on your return. Having all of the income from one state allows you to file as a full-year resident in that state even though you moved: all of your income was produced while you were living there.
There is a caveat: Technically, the correct procedure is to split the interest income proportionately between the two states. Thus, if the interest income is considerable, to the point that you could have a tax liability in the second state if you report the portion of interest that was generated after moving there, then you will need to go through the exercise of filing both state part-year returns and allocating the correct amount of interest to the second state. It's just not likely that you will have enough interest income for this to be necessary.
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